Two Optimized Symmetric Eight-Step Implicit Methods for Initial-Value Problems with Oscillating Solutions
For researchers solving oscillatory IVPs (e.g., quantum mechanics, orbital dynamics), this provides a more efficient numerical method, though the improvement is incremental over existing optimized methods.
The authors developed two optimized eight-step symmetric implicit methods for solving initial-value problems with oscillating solutions, including the radial Schrödinger equation. The method with infinite phase-lag order was the most efficient among all compared methods across all tested problems.
In this paper we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schrödinger equation with the use of the Woods-Saxon potential. They can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the two new methods to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.